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NCERT Class XI Mathematics - Straight Lines - Solutions

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Question : 58
Total: 74
Find perpendicular distance from the origin of the line joining the points (cosθ, sinθ) and (cosϕ, sinϕ).
Solution:  
Let the points be A (cosθ, sinθ) and B(cosϕ, sinϕ)
Equation of the line AB is
y - sin θ =
sinϕ−sinθ
cosϕ−cosϕ
 (x - cos θ)
⇒ y - sin θ =
2cos(
θ+ϕ
2
)sin(
ϕ−θ
2
)
−2sin(
ϕ+θ
2
)sin(
ϕ−θ
2
)
(x−cosθ)

⇒ (y - sin θ) = −
cos(
θ+ϕ
2
)
sin(
θ+ϕ
2
)
(x−cosθ)
⇒ y sin (
θ+ϕ
2
) - sin θ sin (
θ+ϕ
2
) = - x cos (
θ+ϕ
2
) + cos θ . cos (
θ+ϕ
2
)
=
=xcos(
θ+ϕ
2
) + y sin (
θ+ϕ
2
) - [cosθcos(
θ+ϕ
2
)+sinθ.sin(
θ+ϕ
2
)]=0

⇒ x cos (
θ+ϕ
2
) + y sin (
θ+ϕ
2
) - cos (θ−(
θ−ϕ
2
)) = 0
[By using cos A·cosB + sinA·sinB = cos (A – B)]
⇒ x cos (
θ+ϕ
2
) + y sin (
θ+ϕ
2
) - cos (
θ−ϕ
2
) = 0
Now distance of the above line from the origin
=
|0+0−cos(
θ−ϕ
2
)|
√cos2(
θ+ϕ
2
)+sin2(
θ+ϕ
2
)
= |cos(
θ−ϕ
2
)|.
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